Question

Math help
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Answer 1

Based on the given graph, we can determine the solutions to the system of equations y = f(x) and y = g(x), as well as the solutions to the equation f(x) = g(x).

To find the solutions to the system of equations y = f(x) and y = g(x), we need to identify the points where the two graphs intersect. From the graph, it appears that the two graphs intersect at x = 2 and x = 4.

Therefore, the solutions to the system of equations y = f(x) and y = g(x) are (2, 3) and (4, 8).

To find the solutions to the equation f(x) = g(x), we need to find the points where the y-values of both graphs are equal. From the graph, it appears that the y-values are equal at x = 3.

Therefore, the solution to the equation f(x) = g(x) is x = 3.

To summarize:

- The solutions to the system of equations y = f(x) and y = g(x) are (2, 3) and (4, 8).

- The solution to the equation f(x) = g(x) is x = 3.

Please note that without a clear visual representation of the graphs and the points, it is important to verify the accuracy of the solutions by checking the equations algebraically or referring to additional information.

If you have any further questions or need additional clarification, feel free to ask!

Answer 2

The solutions to the system of equations y = f(x) and y = g(x) are (0, 7) and (2, 4).

The solutions to the equation f(x) = g(x) are 0 and 4.

Explanation:

To find the solutions to the system of equations y = f(x) and y = g(x), we need to determine the values of x and y that satisfy both equations simultaneously. In this case, we're looking for the points where the graphs of y = f(x) and y = g(x) intersect.

From observation of the given graph, the points of intersection are:

• (0, 7) and (2, 4)

The solutions to f(x) = g(x) are the values of x at which the two functions are equal to each other. Therefore, we are looking for the x-values of the points of intersection of the two graphs, which are:

• x = 0 and x = 2

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Additional notes

Both answers can be proved algebraically.

Set f(x) equal to g(x):

`\begin{aligned}f(x)&=g(x)\\\\4(2)^(-x)+3&=8-2^x\end{aligned}`

Apply exponent rules to the left side of the equation:

`4(2^x)^(-1)+3=8-2^x`

Let
`u=2^x`:

`4(u)^(-1)+3=8-u`

Solve for u:

`\begin{aligned}4(u)^(-1)+3&=8-u\\\\(4)/(u)+3&=8-u\\\\(4)/(u)+u&=5\\\\4+u^2&=5u\\\\u^2-5u+4&=0\\\\u^2-u-4u+4&=0\\\\u(u-1)-4(u-1)&=0\\\\(u-1)(u-4)&=0\\\\u-1&=0\implies u=1\\u-4&=0\implies u=4\end{aligned}`

Substitute back in
`u=2^x` and solve for x:

`2^x=1 \implies x=0`

`2^x=4 \implies 2^x=2^2 \implies x=2`

Therefore, the solutions to the equation f(x) = g(x) are x = 0 and x = 2.

To find the y-coordinates, substitute the found x-values into one of the functions:

`\begin{aligned}x=0 \implies g(0)&=8-2^0\\g(0)&=8-1\\g(0)&=7\end{aligned}`

`\begin{aligned}x=2 \implies g(2)&=8-2^2\\g(2)&=8-4\\g(2)&=4\end{aligned}`

Therefore, the points of intersection of the two functions are (0, 7) and (2, 4).